Geodesics on the upper half-plane – parametrization

The last note ended with the following problem:

Thus: geodesics are circles. Or better: straight lines are circles! In fact: half-circles, because their centers are on the x-axis, and our arena is only upper half-plane.

Except that we have missed some solutions. In Conformally Euclidean geometry of the upper half-plane there was a sentence:

In the next note we will start calculating “straight lines”, or “geodesics” of our geometry. Some of them are almost evident candidates: vertical lines, perpendicular to the real line. But what about the other ones?

Well, we have these other ones, but how the vertical lines fit our reasoning above?

If \gamma(t)=(x(t),y(t)) is a vertical line, then x(t)=x_0, therefore \dot{x}=0. In Geodesics on the upper half-plane – Part 2 circles we arrived at equations

(1)   \begin{equation*} (\dot{\gamma}(t),K_1(\gamma(t)))=\frac{\dot{x}}{y^2(t)}=\mathrm{const},\end{equation*}

(2)   \begin{equation*} (\dot{\gamma}(t),K_2(\gamma(t)))=\frac{\dot{x}(t)x(t)+\dot{y}(t)y(t)}{y^2(t)}=\mathrm{const},\end{equation*}

and took the ratio of the second to the first. But for vertical lines taking the ratio is not allowed. The first equation is satisfied with the constant on the right hand side equal to zero. The second equation reduces to

(3)   \begin{equation*}(\dot{\gamma}(t),K_2(\gamma(t)))=\frac{\dot{y}(t)}{y(t)}=\mathrm{const}.\end{equation*}

If we now recall Eq. (2) from Conformally Euclidean geometry of the upper half-plane :

(4)   \begin{equation*}ds=\sqrt{\frac{dx^2+dy^2}{y^2}}=\sqrt{\frac{\dot{x}^2+\dot{y}^2}{y^2}}dt,\end{equation*}

we see that ds and dt on the vertical line must be proportional:

(5)   \begin{equation*}s=ct+s_0.\end{equation*}

Whenever this last equation holds, one says that t is an “affine parameter”: it is proportional to the arc length, possibly translated. In fact that is part of the definition of the geodesic that enters the “Noether’s theorem” that we are using. Usually we choose the proportionality constant equal to one.

Of course we can use Eq. (3) in order to determine the parameter t. Choosing the constant equal to 1, we have

(6)   \begin{equation*}\frac{dy}{y}=dt,\end{equation*}


(7)   \begin{equation*}\log y=t+t_0.\end{equation*}

To my surprise Bjab has discovered this all by himself – see the discussion under the previous post

Therefore, when discussing geodesics, we will assume that they are always parametrized by their arc length or, in other words, that the tangent vector \dot{\gamma}(t) is of unit length. In our case that is equivalent to

(8)   \begin{equation*}(\dot{\gamma}(s),\dot{\gamma}(s))_{\gamma(s)}=\frac{\dot{x}(s)^2+\dot{y}(s)^2}{y(s)^2}=1.\end{equation*}

Remark: There are many important examples of pseudo-Riemannian metrics, that are not positive definite. In such a case “length square” along a geodesic line is normalized to +1 or -1 or 0, so that there are three kinds of geodesics. In physics this happens for space-time metrics with Minkowski signature, and in multi-dimensional Kaluza-Klein theories. We will discuss another such case in the following posts.

We can now use these insights also in the case of circular geodesics. Before we have taken the ratio of two “conservation law” equations (1,2). Now, that we know we have a circle, we can write circle equation

(9)   \begin{eqnarray*}x&=&r\cos (\phi)+x_0,\\ y&=&r\sin( \phi),\label{eq:yr}\end{eqnarray*}

where \phi is some function of the arc length parameter s, and substitute into Eqs. (1,2).

We have

(10)   \begin{eqnarray*}\dot{x}&=&-r\sin (\phi)\, \dot{\phi}\\ \dot{y}&=&r\cos (\phi)\,\dot{\phi},\label{eq:dy}\end{eqnarray*}

therefore Eq. (8) reduces to

(11)   \begin{equation*} \frac{\dot{\phi}}{\sin\phi}=\pm 1.\end{equation*}

It is now a straightforward exercise to verify that with Eqs. (911) equations (1,2) are satisfied automatically.

Now that we know geodesics on the upper half-plane, we can draw them for pleasure. There is a famous pattern known as Dedekind’s tesselation

Dedekind tessellation

I was able to reproduce a part of it, namely the part described on p. 3 in the paper on SL(2,Z) by Keith Conrad

SL(2,Z) tessellation

But I would like to be able to reproduce the pretty image from website of Jerzy Kocik from Southern Illinois University

Dedekind tessellation Click on the image to view it full size

And I do not know yet how to do it.

Update: After several hours I managed to produce this:

My poor version – very primitive

Prime numbers are the dwellings of the mystics

In Cayley transform for Easter we used the Cayley transform to create a pattern on the Poincaré disk:

Gaussian fractions on the disk.

Which reminded me of the Ulam spiral

The Ulam spiral or prime spiral (in other languages also called the Ulam cloth) is a graphical depiction of the set of prime numbers, devised by mathematician Stanislaw Ulam in 1963 and popularized in Martin Gardner’s Mathematical Games column in Scientific American a short time later.[1] It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.

Ulam spiral

The organization of prime numbers is somewhat unpredictable, so it was kind of surprise for Ulam to find patterns in his collecting consecutive prime numbers into a spiral.

So, I used Cayley transform and applied it to fractions (m+in)/(p+iq) where m,n,p,q are prime numbers such that the fraction is in the upper half-plane. I restricted primes to be at most 23, included their negatives, and added 0 and 1 to the table.

Here is the resulting pattern:

Pattern of Cayley transformed fractions made of Gaussian integers. Click on the image to open a 1280×1281 magnification

Our eyes can see all kind of patterns there. I have no explanation for these patterns.

Recreations with Cayley transform

What can we do with Cayley transform? We can produce interesting pictures. Here are two such pictures:

Gaussian integers with Cayley transform
Gaussian integers on the Poincaré disk

In fact in both pictures we have the same pattern of dots, but they are organized differently by coloring.

How are these images produced? They are produced using Gaussian integers and Cayley transform.

Cayley transform we know from the previous post Cayley transform for Easter. It is the same as in Wikipedia , where it is defined as

(1)   \begin{equation*}z'=f(z)=\frac{z-i}{z+i}.\end{equation*}

It maps complex upper half-plane \mathbb{H}, the set of all complex numbers with positive imaginary part, onto the interior D of the unit disk. The real axis is mapped onto the unit circle, minus the point z'=1. The inverse Cayley transform maps 1 to infinity.

Gaussian integers are also explained in Wikipedia: \url{}

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. ….

We want our Gaussian integers to be in \mathbb{H}, or on the real line, so we take the integer defining the imaginary part to be nonnegative. To produce images above I took Gaussian integers of the form Z=m+in with m varying from -100 to 100, and n varying from 0 to 100. To each such z I apply the Cayley transform and plot the point f(z).

At the end I rotate the images 90 degrees clockwise, so that the neighborhood of z'=1 is at the bottom. It looks for me more interesting this way.

The colors of the points are constant, either along increasing m or along the increasing n.

Notice that the line with the same color in the first picture look like hyperbolic straight lines – they are circle segments perpendicular to the boundary.

We will need to understand why is it so?